Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Online Graph Filtering Over Expanding Graphs

Bishwadeep Das, Elvin Isufi

TL;DR

The paper tackles graph signal processing on networks that grow over time by introducing online FIR graph filters that adapt to expanding graphs under both deterministic and stochastic attachment models. It develops a deterministic online graph filtering framework (D-OGF) with projected gradient updates and sublinear regret bounds, and extends to stochastic settings with heuristic (S-OGF) and adaptive (Ada-OGF) attachment models, including a prediction-correction variant (PC-OGF). The methods are validated on synthetic and real data (e.g., Movielens and COVID-19) and show competitive or superior performance to baselines such as pre-trained filters and kernel methods, while providing insight into the role of the growth process in regret and learning dynamics. The work demonstrates practical online inference for signals on expanding graphs and lays groundwork for future joint topology and filter learning, spatio-temporal extensions, and distributed implementations.

Abstract

Graph filters are a staple tool for processing signals over graphs in a multitude of downstream tasks. However, they are commonly designed for graphs with a fixed number of nodes, despite real-world networks typically grow over time. This topological evolution is often known up to a stochastic model, thus, making conventional graph filters ill-equipped to withstand such topological changes, their uncertainty, as well as the dynamic nature of the incoming data. To tackle these issues, we propose an online graph filtering framework by relying on online learning principles. We design filters for scenarios where the topology is both known and unknown, including a learner adaptive to such evolution. We conduct a regret analysis to highlight the role played by the different components such as the online algorithm, the filter order, and the growing graph model. Numerical experiments with synthetic and real data corroborate the proposed approach for graph signal inference tasks and show a competitive performance w.r.t. baselines and state-of-the-art alternatives.

Online Graph Filtering Over Expanding Graphs

TL;DR

The paper tackles graph signal processing on networks that grow over time by introducing online FIR graph filters that adapt to expanding graphs under both deterministic and stochastic attachment models. It develops a deterministic online graph filtering framework (D-OGF) with projected gradient updates and sublinear regret bounds, and extends to stochastic settings with heuristic (S-OGF) and adaptive (Ada-OGF) attachment models, including a prediction-correction variant (PC-OGF). The methods are validated on synthetic and real data (e.g., Movielens and COVID-19) and show competitive or superior performance to baselines such as pre-trained filters and kernel methods, while providing insight into the role of the growth process in regret and learning dynamics. The work demonstrates practical online inference for signals on expanding graphs and lays groundwork for future joint topology and filter learning, spatio-temporal extensions, and distributed implementations.

Abstract

Graph filters are a staple tool for processing signals over graphs in a multitude of downstream tasks. However, they are commonly designed for graphs with a fixed number of nodes, despite real-world networks typically grow over time. This topological evolution is often known up to a stochastic model, thus, making conventional graph filters ill-equipped to withstand such topological changes, their uncertainty, as well as the dynamic nature of the incoming data. To tackle these issues, we propose an online graph filtering framework by relying on online learning principles. We design filters for scenarios where the topology is both known and unknown, including a learner adaptive to such evolution. We conduct a regret analysis to highlight the role played by the different components such as the online algorithm, the filter order, and the growing graph model. Numerical experiments with synthetic and real data corroborate the proposed approach for graph signal inference tasks and show a competitive performance w.r.t. baselines and state-of-the-art alternatives.
Paper Structure (17 sections, 7 theorems, 62 equations, 6 figures, 2 tables, 4 algorithms)

This paper contains 17 sections, 7 theorems, 62 equations, 6 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Filtering over Expanding Graphs
  4. Online Filter Learning
  5. Deterministic Online Filtering
  6. Stochastic online filtering
  7. Heuristic Stochastic Online Filtering
  8. Adaptive Stochastic Online Filtering
  9. Numerical Experiments
  10. Experimental Setup
  11. Performance Comparison
  12. Analysis of Online Methods
  13. Conclusion
  14. Proof of Theorem \ref{['Prop 2']}
  15. Proof of Corollary \ref{['Corollary2']}
  16. ...and 2 more sections

Key Result

Proposition 1

Consider a sequence of Lipschitz losses $\{l_t({\mathbf h},x_{t})\}$ with Lipschitz constants $L_d$, $[cf.~instant loss]$ and a learning rate $\eta$$[cf. online update]$. Let also Assumptions assumption1-assumption4 hold. The normalized static regret $R_{T}({\mathbf h}^{\star})$ for the online algor with $L_d=RC+2\mu H$ where $||{\mathbf A}_{x,t-1}^{\top}{\mathbf a}_t||_2\leq C$.

Figures (6)

  • Figure 1: Online filter learning process at time $t$ through the addition of node $v_t$ with signal $x_t$ and the update of the filter ${\mathbf h}(t)$. (Left) node $v_t$ attaches to the previous graph ${\mathcal{G}}_{t-1}$ forming ${\mathcal{G}}_t$; the edges in blue denote the existing edges while those in orange denote the edges formed by the incoming node; (Centre) Signal $x_t$ is predicted, then the true value $x_t$ is revealed, and the filter parameter ${\mathbf h}(t)$ is updated from ${\mathbf h}(t-1)$; (Right) the next node $v_{t+1}$ attaches to ${\mathcal{G}}_{t}$.
  • Figure 2: Box plot of the squared errors for each method in the Movielens data-set.
  • Figure 3: (Left) Box plot of the squared errors across all data points over the six days. (Right) Box plot on the right zoomed in to highlight differences between all methods.
  • Figure 4: RNMSE for different values of filter order $K$ for (left) Filter, (centre) WMean, and (right) Kernel data, respectively.
  • Figure 5: Evolution of the normalized cumulative regret for S-OGF and Ada-OGF for the synthetic (left) Filter, (center) WMean and (right) Kernel data for $T=800$, $T$ and $400$ incoming nodes, respectively.
  • ...and 1 more figures

Theorems & Definitions (9)

  • Remark 1
  • Proposition 1
  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Remark 2
  • Lemma 1
  • Lemma 2
  • Lemma 3